Quantization of Maxwell’s equations

Прохоров Лев Васильевич

Boris N. Chichkov

Quantum theory of photons based on the first quantization technique, similar to that used by Schrödinger in the formulation of quantum mechanics, is considered. First, scalar quantum mechanics of photons operating with the photon wave functions is discussed. Using the first quantization, the wave equation, the Schrödinger-like equations, and the Dirac equation for photons are derived. Then, vector quantum mechanics of photons is introduced, which defines the electromagnetic vector fields. Using the first quantization, the Maxwell equations for photons in a magneto-dielectric medium are obtained. Because the photon’s electric and magnetic fields satisfy the Maxwell equations, all that is known about the classical optical fields can be directly transferred to photons demonstrating their quantum diversity. Relationships between the scalar and vector quantum mechanics of photons and between the Dirac and Maxwell equations are analyzed. To describe the propagation of photons in dispersive media, modified Maxwell equations are introduced.

Weng Cho Chew, Dong-Yeop Na, Peter A. Bermel +3 more

We present a succinct way to quantize Maxwell's equations. We begin by discussing the random nature of quantum observables. Then we present the quantum Maxwell's equations and give their physical meanings. Due to the mathematical homomorphism between the classical and quantum cases [1], the derivation of quantum Maxwell's equations can be simplified. First, one needs only to verify that the classical equations of motion are derivable from a Hamiltonian by energy-conservation arguments. Then the quantum equations of motion follow due to homomorphism, which applies to sum-separable Hamiltonians. Hence, we first show the derivation of classical Maxwell's equations using Hamiltonian theory. Then the derivation of the quantum Maxwell's equations follows in an analogous fashion. Finally, we apply this quantization procedure to the dispersive medium case. (This article is written for the classical electromagnetic community assuming little knowledge of quantum theory, an introduction of which can be found in [2, Lecs. 38 and 39].)

Galina A. Nemova

Thomas A. Garrity

: Our goal is to quantize Maxwell's equations, leading to a natural interpretation of light as being composed of photons. The key is that the quantization of harmonic oscillators (or, more prosaically, the mathematics of springs) is critical to understanding light. In particular, we show that the possible energies of light form a discrete set, linked to the classical frequency, giving us an interpretation for the photoelectric effect. Our Approach From Einstein's explanation of the photoelectric effect, light seems to be made up of photons, which in turn are somewhat particle-like. Can we start with the classical description of light as an electromagnetic wave solution to Maxwell's equations when there are no charges or currents, quantize, and then get something that can be identified with photons? That is the goal of this chapter. (As in the previous chapter, while all of this is standard, we will be following the outline given in chapter 2 of [41].) Maxwell's equations are linear. We will concentrate on the ‘monochromatic’ solutions, those with fixed frequency and direction. We will see that the Hamiltonians of these monochromatic solutions will have the same mathematical structure as a Hamiltonian for a harmonic oscillator. We know from the last chapter how to quantize the harmonic oscillator. More importantly, we saw that there are discrete eigenvalues and eigenvectors for the corresponding Hamiltonian operator. When we quantize the Hamiltonians for the monochromatic waves in an analogous fashion, we will again have discrete eigenvalues and eigenvectors. It is these eigenvectors that we can interpret as photons.

Milan Perkovac